Notation

There are many notations used for the inverse trigonometric functions. The notations sin−1 (x), cos−1 (x), tan−1 (x), etc. are often used, but this convention logically conflicts with the common semantics for expressions like sin2 (x), which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse and compositional inverse. Another convention used by some authors[2] is to use a majuscule (capital/upper-case) first letter along with a −1 superscript, e.g., Sin−1 (x), Cos−1 (x), etc., which avoids confusing them with the multiplicative inverse, which should be represented by sin−1 (x), cos−1 (x), etc. Yet another convention is to use an arc- prefix, so that the confusion with the −1 superscript is resolved completely, e.g., arcsin (x), arccos (x), etc. This convention is used throughout the article. In computer programming languages the inverse trigonometric functions are usually called asin, acos, atan.

Etymology of the arc- prefix

When measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Thus, in the unit circle, "the arc whose cosine is x" is the same as "the angle whose cosine is x", because the measure of the length of the arc of the circle in radii is the same as the measurement of the angle in radians.

Principal values

Since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions. Therefore the ranges of the inverse functions are proper subsets of the domains of the original functions

For example, using function in the sense of multivalued functions, just as the square root function y = √x could be defined from that y2 = x, the function y = arcsin(x) is defined so that sin(y) = x. There are multiple numbers y such that sin(y) = x; for example, sin(0) = 0, but also sin() = 0, sin(2) = 0, etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = , arcsin(0) = 2, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x) will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.

Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1, and another side of length x (any real number between 0 and 1), then applying the Pythagorean theorem and definitions the trigonometric ratios. Purely algebraic derivations are longer.

\theta

\sin \theta

\cos \theta

\tan \theta

Diagram

\arcsin x

\sin (\arcsin x) = x

\cos (\arcsin x) = \sqrt{1-x^2}

\tan (\arcsin x) = \frac{x}{\sqrt{1-x^2}}

\arccos x

\sin (\arccos x) = \sqrt{1-x^2}

\cos (\arccos x) = x

\tan (\arccos x) = \frac{\sqrt{1-x^2}}{x}

\arctan x

\sin (\arctan x) = \frac{x}{\sqrt{1+x^2}}

\cos (\arctan x) = \frac{1}{\sqrt{1+x^2}}

\tan (\arctan x) = x

\arccot x

\sin (\arccot x) = \frac{1}{\sqrt{1+x^2}}

\cos (\arccot x) = \frac{x}{\sqrt{1+x^2}}

\tan (\arccot x) = \frac{1}{x}

\arcsec x

\sin (\arcsec x) = \frac{\sqrt{x^2-1}}{x}

\cos (\arcsec x) = \frac{1}{x}

\tan (\arcsec x) = \sqrt{x^2-1}

\arccsc x

\sin (\arccsc x) = \frac{1}{x}

\cos (\arccsc x) = \frac{\sqrt{x^2-1}}{x}

\tan (\arccsc x) = \frac{1}{\sqrt{x^2-1}}

General solutions

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2. Sine and cosecant begin their period at 2k − /2 (where k is an integer), finish it at 2k + /2, and then reverse themselves over 2k + /2 to 2k + 3/2. Cosine and secant begin their period at 2k, finish it at 2k + , and then reverse themselves over 2k + to 2k + 2. Tangent begins its period at 2k − /2, finishes it at 2k + /2, and then repeats it (forward) over 2k + /2 to 2k + 3/2. Cotangent begins its period at 2k, finishes it at 2k + , and then repeats it (forward) over 2k + to 2k + 2.

This periodicity is reflected in the general inverses where k is some integer:

Extension to complex plane

Since the inverse trigonometric functions are analytic functions, they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points. One possible way of defining the extensions is:

\arctan z = \int_0^z \frac{d x}{1 + x^2} \quad z \neq -i, +i \,

where the part of the imaginary axis which does not lie strictly between −i and +i is the cut between the principal sheet and other sheets;

\arcsin z = \arctan \frac{z}{\sqrt{1 - z^2}} \quad z \neq -1, +1 \,

where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the cut between the principal sheet of arcsin and other sheets;

\arccos z = \frac{\pi}{2} - \arcsin z \quad z \neq -1, +1 \,

which has the same cut as arcsin;

\arccot z = \frac{\pi}{2} - \arctan z \quad z \neq -i, +i \,

which has the same cut as arctan;

\arcsec z = \arccos \frac{1}{z} \quad z \neq -1, 0, +1 \,

where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;

Continued fractions for arctangent

\arctan z=\cfrac{z} {1+\cfrac{(1z)^2} {3-1z^2+\cfrac{(3z)^2} {5-3z^2+\cfrac{(5z)^2} {7-5z^2+\cfrac{(7z)^2} {9-7z^2+\ddots}}}}}
=\cfrac{z} {1+\cfrac{(1z)^2} {3+\cfrac{(2z)^2} {5+\cfrac{(3z)^2} {7+\cfrac{(4z)^2} {9+\ddots\,}}}}}\,
The second of these is valid in the cut complex plane. There are two cuts, from −i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just (nz)2, with each perfect square appearing once. The first was developed by Leonhard Euler; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series.

Two-argument variant of arctangent

The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of (−, ]. In other words, atan2(y, x) is the angle between the positive x-axis of a plane and the point (x, y) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane, y

In terms of the standard arctan function, that is with range of (−/2, /2), it can be expressed as follows:

provided that either x > 0 or y ≠ 0. However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.

The above argument order (y, x) seems to be the most common, and in particular is used in ISO standards such as the C programming language, but a few authors may use the opposite convention (x, y) so some caution is warranted. These variations are detailed at Atan2.

Arctangent function with location parameter

In many applications the solution y of the equation x=\tan y is to come as close as possible to a given value -\infty. The adequate solution is produced by the parameter modified arctangent function

Application: finding the angle of a right triangle

Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine, for example, it follows that

Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem: a^2+b^2=h^2 where h is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.

Practical considerations

For angles near 0 and , arccosine is ill-conditioned and will thus calculate the angle with reduced accuracy in a computer implementation (due to the limited number of digits). Similarly, arcsine is inaccurate for angles near −/2 and /2. To achieve full accuracy for all angles, arctangent or atan2 should be used for the implementation.

References

External links

MathWorld.

MathWorld.

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